WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 30.5 s] (2) CpxRelTRS (3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (4) TRS for Loop Detection (5) DecreasingLoopProof [LOWER BOUND(ID), 2 ms] (6) BEST (7) proven lower bound (8) LowerBoundPropagationProof [FINISHED, 0 ms] (9) BOUNDS(n^1, INF) (10) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__AND(tt, z0) -> c(MARK(z0)) A__AND(z0, z1) -> c1 A__ISNATILIST(z0) -> c2(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c3 A__ISNATILIST(cons(z0, z1)) -> c4(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(cons(z0, z1)) -> c5(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATILIST(z0) -> c6 A__ISNAT(0) -> c7 A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(length(z0)) -> c9(A__ISNATLIST(z0)) A__ISNAT(z0) -> c10 A__ISNATLIST(nil) -> c11 A__ISNATLIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c13(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__ISNATLIST(take(z0, z1)) -> c14(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(take(z0, z1)) -> c15(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATLIST(z0) -> c16 A__ZEROS -> c17 A__ZEROS -> c18 A__TAKE(0, z0) -> c19(A__UTAKE1(a__isNatIList(z0)), A__ISNATILIST(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c20(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__ISNAT(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c21(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNAT(z1)) A__TAKE(s(z0), cons(z1, z2)) -> c22(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNATILIST(z2)) A__TAKE(z0, z1) -> c23 A__UTAKE1(tt) -> c24 A__UTAKE1(z0) -> c25 A__UTAKE2(tt, z0, z1, z2) -> c26(MARK(z1)) A__UTAKE2(z0, z1, z2, z3) -> c27 A__LENGTH(cons(z0, z1)) -> c28(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c29(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c30 A__ULENGTH(tt, z0) -> c31(A__LENGTH(mark(z0)), MARK(z0)) A__ULENGTH(z0, z1) -> c32 MARK(and(z0, z1)) -> c33(A__AND(mark(z0), mark(z1)), MARK(z0)) MARK(and(z0, z1)) -> c34(A__AND(mark(z0), mark(z1)), MARK(z1)) MARK(isNatIList(z0)) -> c35(A__ISNATILIST(z0)) MARK(isNatList(z0)) -> c36(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c37(A__ISNAT(z0)) MARK(length(z0)) -> c38(A__LENGTH(mark(z0)), MARK(z0)) MARK(zeros) -> c39(A__ZEROS) MARK(take(z0, z1)) -> c40(A__TAKE(mark(z0), mark(z1)), MARK(z0)) MARK(take(z0, z1)) -> c41(A__TAKE(mark(z0), mark(z1)), MARK(z1)) MARK(uTake1(z0)) -> c42(A__UTAKE1(mark(z0)), MARK(z0)) MARK(uTake2(z0, z1, z2, z3)) -> c43(A__UTAKE2(mark(z0), z1, z2, z3), MARK(z0)) MARK(uLength(z0, z1)) -> c44(A__ULENGTH(mark(z0), z1), MARK(z0)) MARK(tt) -> c45 MARK(0) -> c46 MARK(s(z0)) -> c47(MARK(z0)) MARK(cons(z0, z1)) -> c48(MARK(z0)) MARK(nil) -> c49 The (relative) TRS S consists of the following rules: a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNat(0) -> tt a__isNat(s(z0)) -> a__isNat(z0) a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(z0) -> isNat(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatList(z1)) a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatList(z0) -> isNatList(z0) a__zeros -> cons(0, zeros) a__zeros -> zeros a__take(0, z0) -> a__uTake1(a__isNatIList(z0)) a__take(s(z0), cons(z1, z2)) -> a__uTake2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2) a__take(z0, z1) -> take(z0, z1) a__uTake1(tt) -> nil a__uTake1(z0) -> uTake1(z0) a__uTake2(tt, z0, z1, z2) -> cons(mark(z1), take(z0, z2)) a__uTake2(z0, z1, z2, z3) -> uTake2(z0, z1, z2, z3) a__length(cons(z0, z1)) -> a__uLength(a__and(a__isNat(z0), a__isNatList(z1)), z1) a__length(z0) -> length(z0) a__uLength(tt, z0) -> s(a__length(mark(z0))) a__uLength(z0, z1) -> uLength(z0, z1) mark(and(z0, z1)) -> a__and(mark(z0), mark(z1)) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNat(z0)) -> a__isNat(z0) mark(length(z0)) -> a__length(mark(z0)) mark(zeros) -> a__zeros mark(take(z0, z1)) -> a__take(mark(z0), mark(z1)) mark(uTake1(z0)) -> a__uTake1(mark(z0)) mark(uTake2(z0, z1, z2, z3)) -> a__uTake2(mark(z0), z1, z2, z3) mark(uLength(z0, z1)) -> a__uLength(mark(z0), z1) mark(tt) -> tt mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__AND(tt, z0) -> c(MARK(z0)) A__AND(z0, z1) -> c1 A__ISNATILIST(z0) -> c2(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c3 A__ISNATILIST(cons(z0, z1)) -> c4(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(cons(z0, z1)) -> c5(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATILIST(z0) -> c6 A__ISNAT(0) -> c7 A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(length(z0)) -> c9(A__ISNATLIST(z0)) A__ISNAT(z0) -> c10 A__ISNATLIST(nil) -> c11 A__ISNATLIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c13(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__ISNATLIST(take(z0, z1)) -> c14(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(take(z0, z1)) -> c15(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATLIST(z0) -> c16 A__ZEROS -> c17 A__ZEROS -> c18 A__TAKE(0, z0) -> c19(A__UTAKE1(a__isNatIList(z0)), A__ISNATILIST(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c20(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__ISNAT(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c21(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNAT(z1)) A__TAKE(s(z0), cons(z1, z2)) -> c22(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNATILIST(z2)) A__TAKE(z0, z1) -> c23 A__UTAKE1(tt) -> c24 A__UTAKE1(z0) -> c25 A__UTAKE2(tt, z0, z1, z2) -> c26(MARK(z1)) A__UTAKE2(z0, z1, z2, z3) -> c27 A__LENGTH(cons(z0, z1)) -> c28(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c29(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c30 A__ULENGTH(tt, z0) -> c31(A__LENGTH(mark(z0)), MARK(z0)) A__ULENGTH(z0, z1) -> c32 MARK(and(z0, z1)) -> c33(A__AND(mark(z0), mark(z1)), MARK(z0)) MARK(and(z0, z1)) -> c34(A__AND(mark(z0), mark(z1)), MARK(z1)) MARK(isNatIList(z0)) -> c35(A__ISNATILIST(z0)) MARK(isNatList(z0)) -> c36(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c37(A__ISNAT(z0)) MARK(length(z0)) -> c38(A__LENGTH(mark(z0)), MARK(z0)) MARK(zeros) -> c39(A__ZEROS) MARK(take(z0, z1)) -> c40(A__TAKE(mark(z0), mark(z1)), MARK(z0)) MARK(take(z0, z1)) -> c41(A__TAKE(mark(z0), mark(z1)), MARK(z1)) MARK(uTake1(z0)) -> c42(A__UTAKE1(mark(z0)), MARK(z0)) MARK(uTake2(z0, z1, z2, z3)) -> c43(A__UTAKE2(mark(z0), z1, z2, z3), MARK(z0)) MARK(uLength(z0, z1)) -> c44(A__ULENGTH(mark(z0), z1), MARK(z0)) MARK(tt) -> c45 MARK(0) -> c46 MARK(s(z0)) -> c47(MARK(z0)) MARK(cons(z0, z1)) -> c48(MARK(z0)) MARK(nil) -> c49 The (relative) TRS S consists of the following rules: a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNat(0) -> tt a__isNat(s(z0)) -> a__isNat(z0) a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(z0) -> isNat(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatList(z1)) a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatList(z0) -> isNatList(z0) a__zeros -> cons(0, zeros) a__zeros -> zeros a__take(0, z0) -> a__uTake1(a__isNatIList(z0)) a__take(s(z0), cons(z1, z2)) -> a__uTake2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2) a__take(z0, z1) -> take(z0, z1) a__uTake1(tt) -> nil a__uTake1(z0) -> uTake1(z0) a__uTake2(tt, z0, z1, z2) -> cons(mark(z1), take(z0, z2)) a__uTake2(z0, z1, z2, z3) -> uTake2(z0, z1, z2, z3) a__length(cons(z0, z1)) -> a__uLength(a__and(a__isNat(z0), a__isNatList(z1)), z1) a__length(z0) -> length(z0) a__uLength(tt, z0) -> s(a__length(mark(z0))) a__uLength(z0, z1) -> uLength(z0, z1) mark(and(z0, z1)) -> a__and(mark(z0), mark(z1)) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNat(z0)) -> a__isNat(z0) mark(length(z0)) -> a__length(mark(z0)) mark(zeros) -> a__zeros mark(take(z0, z1)) -> a__take(mark(z0), mark(z1)) mark(uTake1(z0)) -> a__uTake1(mark(z0)) mark(uTake2(z0, z1, z2, z3)) -> a__uTake2(mark(z0), z1, z2, z3) mark(uLength(z0, z1)) -> a__uLength(mark(z0), z1) mark(tt) -> tt mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (4) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__AND(tt, z0) -> c(MARK(z0)) A__AND(z0, z1) -> c1 A__ISNATILIST(z0) -> c2(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c3 A__ISNATILIST(cons(z0, z1)) -> c4(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(cons(z0, z1)) -> c5(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATILIST(z0) -> c6 A__ISNAT(0) -> c7 A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(length(z0)) -> c9(A__ISNATLIST(z0)) A__ISNAT(z0) -> c10 A__ISNATLIST(nil) -> c11 A__ISNATLIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c13(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__ISNATLIST(take(z0, z1)) -> c14(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(take(z0, z1)) -> c15(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATLIST(z0) -> c16 A__ZEROS -> c17 A__ZEROS -> c18 A__TAKE(0, z0) -> c19(A__UTAKE1(a__isNatIList(z0)), A__ISNATILIST(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c20(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__ISNAT(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c21(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNAT(z1)) A__TAKE(s(z0), cons(z1, z2)) -> c22(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNATILIST(z2)) A__TAKE(z0, z1) -> c23 A__UTAKE1(tt) -> c24 A__UTAKE1(z0) -> c25 A__UTAKE2(tt, z0, z1, z2) -> c26(MARK(z1)) A__UTAKE2(z0, z1, z2, z3) -> c27 A__LENGTH(cons(z0, z1)) -> c28(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c29(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c30 A__ULENGTH(tt, z0) -> c31(A__LENGTH(mark(z0)), MARK(z0)) A__ULENGTH(z0, z1) -> c32 MARK(and(z0, z1)) -> c33(A__AND(mark(z0), mark(z1)), MARK(z0)) MARK(and(z0, z1)) -> c34(A__AND(mark(z0), mark(z1)), MARK(z1)) MARK(isNatIList(z0)) -> c35(A__ISNATILIST(z0)) MARK(isNatList(z0)) -> c36(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c37(A__ISNAT(z0)) MARK(length(z0)) -> c38(A__LENGTH(mark(z0)), MARK(z0)) MARK(zeros) -> c39(A__ZEROS) MARK(take(z0, z1)) -> c40(A__TAKE(mark(z0), mark(z1)), MARK(z0)) MARK(take(z0, z1)) -> c41(A__TAKE(mark(z0), mark(z1)), MARK(z1)) MARK(uTake1(z0)) -> c42(A__UTAKE1(mark(z0)), MARK(z0)) MARK(uTake2(z0, z1, z2, z3)) -> c43(A__UTAKE2(mark(z0), z1, z2, z3), MARK(z0)) MARK(uLength(z0, z1)) -> c44(A__ULENGTH(mark(z0), z1), MARK(z0)) MARK(tt) -> c45 MARK(0) -> c46 MARK(s(z0)) -> c47(MARK(z0)) MARK(cons(z0, z1)) -> c48(MARK(z0)) MARK(nil) -> c49 The (relative) TRS S consists of the following rules: a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNat(0) -> tt a__isNat(s(z0)) -> a__isNat(z0) a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(z0) -> isNat(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatList(z1)) a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatList(z0) -> isNatList(z0) a__zeros -> cons(0, zeros) a__zeros -> zeros a__take(0, z0) -> a__uTake1(a__isNatIList(z0)) a__take(s(z0), cons(z1, z2)) -> a__uTake2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2) a__take(z0, z1) -> take(z0, z1) a__uTake1(tt) -> nil a__uTake1(z0) -> uTake1(z0) a__uTake2(tt, z0, z1, z2) -> cons(mark(z1), take(z0, z2)) a__uTake2(z0, z1, z2, z3) -> uTake2(z0, z1, z2, z3) a__length(cons(z0, z1)) -> a__uLength(a__and(a__isNat(z0), a__isNatList(z1)), z1) a__length(z0) -> length(z0) a__uLength(tt, z0) -> s(a__length(mark(z0))) a__uLength(z0, z1) -> uLength(z0, z1) mark(and(z0, z1)) -> a__and(mark(z0), mark(z1)) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNat(z0)) -> a__isNat(z0) mark(length(z0)) -> a__length(mark(z0)) mark(zeros) -> a__zeros mark(take(z0, z1)) -> a__take(mark(z0), mark(z1)) mark(uTake1(z0)) -> a__uTake1(mark(z0)) mark(uTake2(z0, z1, z2, z3)) -> a__uTake2(mark(z0), z1, z2, z3) mark(uLength(z0, z1)) -> a__uLength(mark(z0), z1) mark(tt) -> tt mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (5) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence MARK(and(z0, z1)) ->^+ c33(A__AND(mark(z0), mark(z1)), MARK(z0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [z0 / and(z0, z1)]. The result substitution is [ ]. ---------------------------------------- (6) Complex Obligation (BEST) ---------------------------------------- (7) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__AND(tt, z0) -> c(MARK(z0)) A__AND(z0, z1) -> c1 A__ISNATILIST(z0) -> c2(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c3 A__ISNATILIST(cons(z0, z1)) -> c4(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(cons(z0, z1)) -> c5(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATILIST(z0) -> c6 A__ISNAT(0) -> c7 A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(length(z0)) -> c9(A__ISNATLIST(z0)) A__ISNAT(z0) -> c10 A__ISNATLIST(nil) -> c11 A__ISNATLIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c13(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__ISNATLIST(take(z0, z1)) -> c14(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(take(z0, z1)) -> c15(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATLIST(z0) -> c16 A__ZEROS -> c17 A__ZEROS -> c18 A__TAKE(0, z0) -> c19(A__UTAKE1(a__isNatIList(z0)), A__ISNATILIST(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c20(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__ISNAT(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c21(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNAT(z1)) A__TAKE(s(z0), cons(z1, z2)) -> c22(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNATILIST(z2)) A__TAKE(z0, z1) -> c23 A__UTAKE1(tt) -> c24 A__UTAKE1(z0) -> c25 A__UTAKE2(tt, z0, z1, z2) -> c26(MARK(z1)) A__UTAKE2(z0, z1, z2, z3) -> c27 A__LENGTH(cons(z0, z1)) -> c28(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c29(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c30 A__ULENGTH(tt, z0) -> c31(A__LENGTH(mark(z0)), MARK(z0)) A__ULENGTH(z0, z1) -> c32 MARK(and(z0, z1)) -> c33(A__AND(mark(z0), mark(z1)), MARK(z0)) MARK(and(z0, z1)) -> c34(A__AND(mark(z0), mark(z1)), MARK(z1)) MARK(isNatIList(z0)) -> c35(A__ISNATILIST(z0)) MARK(isNatList(z0)) -> c36(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c37(A__ISNAT(z0)) MARK(length(z0)) -> c38(A__LENGTH(mark(z0)), MARK(z0)) MARK(zeros) -> c39(A__ZEROS) MARK(take(z0, z1)) -> c40(A__TAKE(mark(z0), mark(z1)), MARK(z0)) MARK(take(z0, z1)) -> c41(A__TAKE(mark(z0), mark(z1)), MARK(z1)) MARK(uTake1(z0)) -> c42(A__UTAKE1(mark(z0)), MARK(z0)) MARK(uTake2(z0, z1, z2, z3)) -> c43(A__UTAKE2(mark(z0), z1, z2, z3), MARK(z0)) MARK(uLength(z0, z1)) -> c44(A__ULENGTH(mark(z0), z1), MARK(z0)) MARK(tt) -> c45 MARK(0) -> c46 MARK(s(z0)) -> c47(MARK(z0)) MARK(cons(z0, z1)) -> c48(MARK(z0)) MARK(nil) -> c49 The (relative) TRS S consists of the following rules: a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNat(0) -> tt a__isNat(s(z0)) -> a__isNat(z0) a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(z0) -> isNat(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatList(z1)) a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatList(z0) -> isNatList(z0) a__zeros -> cons(0, zeros) a__zeros -> zeros a__take(0, z0) -> a__uTake1(a__isNatIList(z0)) a__take(s(z0), cons(z1, z2)) -> a__uTake2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2) a__take(z0, z1) -> take(z0, z1) a__uTake1(tt) -> nil a__uTake1(z0) -> uTake1(z0) a__uTake2(tt, z0, z1, z2) -> cons(mark(z1), take(z0, z2)) a__uTake2(z0, z1, z2, z3) -> uTake2(z0, z1, z2, z3) a__length(cons(z0, z1)) -> a__uLength(a__and(a__isNat(z0), a__isNatList(z1)), z1) a__length(z0) -> length(z0) a__uLength(tt, z0) -> s(a__length(mark(z0))) a__uLength(z0, z1) -> uLength(z0, z1) mark(and(z0, z1)) -> a__and(mark(z0), mark(z1)) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNat(z0)) -> a__isNat(z0) mark(length(z0)) -> a__length(mark(z0)) mark(zeros) -> a__zeros mark(take(z0, z1)) -> a__take(mark(z0), mark(z1)) mark(uTake1(z0)) -> a__uTake1(mark(z0)) mark(uTake2(z0, z1, z2, z3)) -> a__uTake2(mark(z0), z1, z2, z3) mark(uLength(z0, z1)) -> a__uLength(mark(z0), z1) mark(tt) -> tt mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (8) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (9) BOUNDS(n^1, INF) ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__AND(tt, z0) -> c(MARK(z0)) A__AND(z0, z1) -> c1 A__ISNATILIST(z0) -> c2(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c3 A__ISNATILIST(cons(z0, z1)) -> c4(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(cons(z0, z1)) -> c5(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATILIST(z0) -> c6 A__ISNAT(0) -> c7 A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(length(z0)) -> c9(A__ISNATLIST(z0)) A__ISNAT(z0) -> c10 A__ISNATLIST(nil) -> c11 A__ISNATLIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c13(A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__ISNATLIST(take(z0, z1)) -> c14(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(take(z0, z1)) -> c15(A__AND(a__isNat(z0), a__isNatIList(z1)), A__ISNATILIST(z1)) A__ISNATLIST(z0) -> c16 A__ZEROS -> c17 A__ZEROS -> c18 A__TAKE(0, z0) -> c19(A__UTAKE1(a__isNatIList(z0)), A__ISNATILIST(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c20(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__ISNAT(z0)) A__TAKE(s(z0), cons(z1, z2)) -> c21(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNAT(z1)) A__TAKE(s(z0), cons(z1, z2)) -> c22(A__UTAKE2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2), A__AND(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), A__AND(a__isNat(z1), a__isNatIList(z2)), A__ISNATILIST(z2)) A__TAKE(z0, z1) -> c23 A__UTAKE1(tt) -> c24 A__UTAKE1(z0) -> c25 A__UTAKE2(tt, z0, z1, z2) -> c26(MARK(z1)) A__UTAKE2(z0, z1, z2, z3) -> c27 A__LENGTH(cons(z0, z1)) -> c28(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c29(A__ULENGTH(a__and(a__isNat(z0), a__isNatList(z1)), z1), A__AND(a__isNat(z0), a__isNatList(z1)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c30 A__ULENGTH(tt, z0) -> c31(A__LENGTH(mark(z0)), MARK(z0)) A__ULENGTH(z0, z1) -> c32 MARK(and(z0, z1)) -> c33(A__AND(mark(z0), mark(z1)), MARK(z0)) MARK(and(z0, z1)) -> c34(A__AND(mark(z0), mark(z1)), MARK(z1)) MARK(isNatIList(z0)) -> c35(A__ISNATILIST(z0)) MARK(isNatList(z0)) -> c36(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c37(A__ISNAT(z0)) MARK(length(z0)) -> c38(A__LENGTH(mark(z0)), MARK(z0)) MARK(zeros) -> c39(A__ZEROS) MARK(take(z0, z1)) -> c40(A__TAKE(mark(z0), mark(z1)), MARK(z0)) MARK(take(z0, z1)) -> c41(A__TAKE(mark(z0), mark(z1)), MARK(z1)) MARK(uTake1(z0)) -> c42(A__UTAKE1(mark(z0)), MARK(z0)) MARK(uTake2(z0, z1, z2, z3)) -> c43(A__UTAKE2(mark(z0), z1, z2, z3), MARK(z0)) MARK(uLength(z0, z1)) -> c44(A__ULENGTH(mark(z0), z1), MARK(z0)) MARK(tt) -> c45 MARK(0) -> c46 MARK(s(z0)) -> c47(MARK(z0)) MARK(cons(z0, z1)) -> c48(MARK(z0)) MARK(nil) -> c49 The (relative) TRS S consists of the following rules: a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNat(0) -> tt a__isNat(s(z0)) -> a__isNat(z0) a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(z0) -> isNat(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), a__isNatList(z1)) a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), a__isNatIList(z1)) a__isNatList(z0) -> isNatList(z0) a__zeros -> cons(0, zeros) a__zeros -> zeros a__take(0, z0) -> a__uTake1(a__isNatIList(z0)) a__take(s(z0), cons(z1, z2)) -> a__uTake2(a__and(a__isNat(z0), a__and(a__isNat(z1), a__isNatIList(z2))), z0, z1, z2) a__take(z0, z1) -> take(z0, z1) a__uTake1(tt) -> nil a__uTake1(z0) -> uTake1(z0) a__uTake2(tt, z0, z1, z2) -> cons(mark(z1), take(z0, z2)) a__uTake2(z0, z1, z2, z3) -> uTake2(z0, z1, z2, z3) a__length(cons(z0, z1)) -> a__uLength(a__and(a__isNat(z0), a__isNatList(z1)), z1) a__length(z0) -> length(z0) a__uLength(tt, z0) -> s(a__length(mark(z0))) a__uLength(z0, z1) -> uLength(z0, z1) mark(and(z0, z1)) -> a__and(mark(z0), mark(z1)) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNat(z0)) -> a__isNat(z0) mark(length(z0)) -> a__length(mark(z0)) mark(zeros) -> a__zeros mark(take(z0, z1)) -> a__take(mark(z0), mark(z1)) mark(uTake1(z0)) -> a__uTake1(mark(z0)) mark(uTake2(z0, z1, z2, z3)) -> a__uTake2(mark(z0), z1, z2, z3) mark(uLength(z0, z1)) -> a__uLength(mark(z0), z1) mark(tt) -> tt mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(nil) -> nil Rewrite Strategy: INNERMOST